Optimal Execution Strategies in Limit Order Books with General Shape Functions

Quantitative Finance (Impact Factor: 0.65). 08/2007; 10(2). DOI: 10.2139/ssrn.1510104
Source: arXiv
We consider optimal execution strategies for block market orders placed in a limit order book (LOB). We build on the resilience model proposed by Obizhaeva and Wang (200516.

Obizhaeva , A and
Wang , J . 2005. Optimal trading strategy and supply/demand dynamics, Preprint Available online at: (accessed 16 February 2009) [CrossRef]View all references) but allow for a general shape of the LOB defined via a given density function. Thus, we can allow for empirically observed LOB shapes and obtain a nonlinear price impact of market orders. We distinguish two possibilities for modelling the resilience of the LOB after a large market order: the exponential recovery of the number of limit orders, i.e. of the volume of the LOB, or the exponential recovery of the bid–ask spread. We consider both of these resilience modes and, in each case, derive explicit optimal execution strategies in discrete time. Applying our results to a block-shaped LOB, we obtain a new closed-form representation for the optimal strategy of a risk-neutral investor, which explicitly solves the recursive scheme given in Obizhaeva and Wang (200516.

Obizhaeva , A and
Wang , J . 2005. Optimal trading strategy and supply/demand dynamics, Preprint Available online at: (accessed 16 February 2009) [CrossRef]View all references). We also provide some evidence for the robustness of optimal strategies with respect to the choice of the shape function and the resilience-type.

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    • "which is nonpositive for nonnegative price S t , hence meaning that due to the price impact, one has to sell. Moreover, in the limiting case where λ → ∞, i.e. the terminal inventory X T is constrained to achieve the target H, then α * ,S is zero: we retrieve the result that the optimal trading rate does not depend on the price process when it is a martingale, see [1], [33]. On the other hand, by applying Itô's formula to (4.2), and using (4.3)-(4.4), "
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    • "For example, we notice that very recently Besson and Lasnier (2015) study the resiliency properties across different European markets and improve the standard market impact model using their resiliency indicator. In addition, we believe that LOB resiliency can also be applied to improve the estimation of the transient or permanent price impact (Bouchaud et al., 2004Bouchaud et al., , 2006 Farmer et al., 2006), to solve the optimal trade execution problem more precisely (Alfonsi et al., 2010; Obizhaeva and Wang, 2013), and to design and calibrate computational models for order-driven markets (Mike and Farmer, 2008; Gu and Zhou, 2009; Li et al., 2014; Sornette, 2014). "
    [Show abstract] [Hide abstract] ABSTRACT: In order-driven markets, limit-order book (LOB) resiliency is an important microscopic indicator of market quality when the order book is hit by a liquidity shock and plays an essential role in the design of optimal submission strategies of large orders. However, the evolutionary behavior of LOB resilience around liquidity shocks is not well understood empirically. Using order flow data sets of Chinese stocks, we quantify and compare the LOB dynamics characterized by the bid-ask spread, the LOB depth and the order intensity surrounding effective market orders with different aggressiveness. We find that traders are more likely to submit effective market orders when the spreads are relatively low, the same-side depth is high, and the opposite-side depth is low. Such phenomenon is especially significant when the initial spread is 1 tick. Although the resiliency patterns show obvious diversity after different types of market orders, the spread and depth can return to the sample average within 20 best limit updates. The price resiliency behavior is dominant after aggressive market orders, while the price continuation behavior is dominant after less-aggressive market orders. Moreover, the effective market orders produce asymmetrical stimulus to limit orders when the initial spreads equal to 1 tick. Under this case, effective buy market orders attract more buy limit orders and effective sell market orders attract more sell limit orders. The resiliency behavior of spread and depth is linked to limit order intensity. Finally, we present applications for high-frequency arbitrage based on LOB resiliency analysis and find that both long and short arbitrage strategies we design can achieve significantly positive returns.
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    • "As recently discussed in Kallsen and Muhle-Karbe [17], the former can be regarded as the high-resilience limit of the latter. Within these two models, most of the existing literature studies the problem of optimally liquidating an exogenously given position by some fixed time horizon (cf., e.g., Almgren and Chriss [2], Almgren [4], Schied and Schöneborn [24], Obizhaeva and Wang [21], Alfonsi et al. [1] and Predoiu et al. [22]). Further work is also devoted to the more involved problem of optimal portfolio choice, cf., e.g., Gârleanu and Pedersen [11], [12], Moreau et al. [19], Guasoni and Weber [14], [15] and Kallsen and Muhle-Karbe [17]. "
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